Extending some functions to functions satisfying condition \((A_3)\).

Abstract

A function \(f:\mathbb R \to \mathbb R\) satisfies the condition \(({\mathcal A}_3)\) if for each real \(r > 0\), for each \(x\) and for each set \(U \ni x\) belonging to the density topology there is an open interval \(I\) such that \(C(f) \supset I \cap U \neq \emptyset \) and \(f(U\cap I) \subset (f(x)-r,f(x)+r)\), where \(C(f)\) denotes the set of all continuity points of \(f\). In this article we investigate the sets \(A\) such that each almost continuous function may be extended from \(A\) to a function having property \(({\mathcal A}_3)\).